MainPage.xaml.cs

using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Runtime.InteropServices.WindowsRuntime;
using Windows.Foundation;
using Windows.Foundation.Collections;
using Windows.UI.Xaml;
using Windows.UI.Xaml.Controls;
using Windows.UI.Xaml.Controls.Primitives;
using Windows.UI.Xaml.Data;
using Windows.UI.Xaml.Input;
using Windows.UI.Xaml.Media;
using Windows.UI.Xaml.Navigation;

using Windows.ApplicationModel.DataTransfer;
using System.Threading.Tasks;
// La plantilla de elemento Página en blanco está documentada en https://go.microsoft.com/fwlink/?LinkId=402352&clcid=0xc0a

namespace ClipboardCSharpUWP
{
    /// <summary>
    /// Página vacía que se puede usar de forma independiente o a la que se puede navegar dentro de un objeto Frame.
    /// </summary>
    public sealed partial class MainPage : Page
    {
        public MainPage()
        {
            this.InitializeComponent();
        }
        public void InputClipboardText(string text)
        {
            DataPackage dataPackage = new DataPackage
            {
                RequestedOperation = DataPackageOperation.Copy
            };
            dataPackage.SetText(text);
            Clipboard.SetContent(dataPackage);
        }
        public static async Task<string> OutputClipboardText()
        {
            DataPackageView dataPackageView = Clipboard.GetContent();

            if (dataPackageView.Contains(StandardDataFormats.Text))
            {
                string text = await dataPackageView.GetTextAsync();
                return text;
            }
            return await Task.FromResult(String.Empty);
        }
        private void BtnCopiar_Click_1(object sender, RoutedEventArgs e)
        {
            InputClipboardText(TxtTexto.Text);
        }

        private async void BtnPegar_Click_1(object sender, RoutedEventArgs e)
        {
            TxtTexto.Text = await OutputClipboardText();
        }
    }
}